MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  arwhoma Structured version   Visualization version   GIF version

Theorem arwhoma 17990
Description: An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
arwrcl.a 𝐴 = (Arrow‘𝐶)
arwhoma.h 𝐻 = (Homa𝐶)
Assertion
Ref Expression
arwhoma (𝐹𝐴𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))

Proof of Theorem arwhoma
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 arwrcl.a . . . . . . 7 𝐴 = (Arrow‘𝐶)
2 arwhoma.h . . . . . . 7 𝐻 = (Homa𝐶)
31, 2arwval 17988 . . . . . 6 𝐴 = ran 𝐻
43eleq2i 2826 . . . . 5 (𝐹𝐴𝐹 ran 𝐻)
54biimpi 215 . . . 4 (𝐹𝐴𝐹 ran 𝐻)
6 eqid 2733 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
71arwrcl 17989 . . . . . 6 (𝐹𝐴𝐶 ∈ Cat)
82, 6, 7homaf 17975 . . . . 5 (𝐹𝐴𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
9 ffn 6713 . . . . 5 (𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V) → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)))
10 fnunirn 7247 . . . . 5 (𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)) → (𝐹 ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻𝑧)))
118, 9, 103syl 18 . . . 4 (𝐹𝐴 → (𝐹 ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻𝑧)))
125, 11mpbid 231 . . 3 (𝐹𝐴 → ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻𝑧))
13 fveq2 6887 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
14 df-ov 7406 . . . . . 6 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
1513, 14eqtr4di 2791 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
1615eleq2d 2820 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹 ∈ (𝐻𝑧) ↔ 𝐹 ∈ (𝑥𝐻𝑦)))
1716rexxp 5839 . . 3 (∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻𝑧) ↔ ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦))
1812, 17sylib 217 . 2 (𝐹𝐴 → ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦))
19 id 22 . . . . 5 (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ (𝑥𝐻𝑦))
202homadm 17985 . . . . . 6 (𝐹 ∈ (𝑥𝐻𝑦) → (doma𝐹) = 𝑥)
212homacd 17986 . . . . . 6 (𝐹 ∈ (𝑥𝐻𝑦) → (coda𝐹) = 𝑦)
2220, 21oveq12d 7421 . . . . 5 (𝐹 ∈ (𝑥𝐻𝑦) → ((doma𝐹)𝐻(coda𝐹)) = (𝑥𝐻𝑦))
2319, 22eleqtrrd 2837 . . . 4 (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
2423rexlimivw 3152 . . 3 (∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
2524rexlimivw 3152 . 2 (∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
2618, 25syl 17 1 (𝐹𝐴𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  wrex 3071  Vcvv 3475  𝒫 cpw 4600  cop 4632   cuni 4906   × cxp 5672  ran crn 5675   Fn wfn 6534  wf 6535  cfv 6539  (class class class)co 7403  Basecbs 17139  domacdoma 17965  codaccoda 17966  Arrowcarw 17967  Homachoma 17968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7406  df-1st 7969  df-2nd 7970  df-doma 17969  df-coda 17970  df-homa 17971  df-arw 17972
This theorem is referenced by:  arwdm  17992  arwcd  17993  arwhom  17996  arwdmcd  17997  coapm  18016
  Copyright terms: Public domain W3C validator